Timeline for Integration on high dimensional sphere
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22, 2012 at 15:36 | answer | added | Igor Rivin | timeline score: 2 | |
| Jul 22, 2012 at 15:25 | comment | added | Andrew | One possible way is to use Monte Carlo method. | |
| Jul 22, 2012 at 15:21 | comment | added | Noam D. Elkies | It's still a nontrivial problem to integrate this numerically. If I did this right it comes down to an elementary multiple of the hyperelliptic integral $$ \int_0^\infty \left[ 1 - \det(1+{\bf M}t^2)^{-1/2} \right] \frac{dt}{t^2}. $$ By the way, the notation $du_1 \cdots du_n$ for the integration element is misleading: it suggests an $n$-dimensional integral, but $\|{\bf U}\|^2 = 1$ is an $n-1$-dimensional manifold. | |
| Jul 22, 2012 at 10:44 | comment | added | David Roberts♦ | You may get a better response at e.g. math.stackexchange.com, as MO is for questions of research interest (see the FAQ for more suggestions). | |
| Jul 22, 2012 at 9:33 | comment | added | Mateusz Wasilewski | You can assume that $\bf{M}$ is diagonal, because orthogonal group acts on a sphere in a measure-preserving way, but that doesn't help, because even in two-dimensional case you end up with an elliptic integral. | |
| Jul 22, 2012 at 7:11 | history | asked | Drinker | CC BY-SA 3.0 |